let n be non empty Nat; :: thesis: for S being non empty non void n PC-correct PCLangSignature

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds

(A \or C) \imp (B \or D) in F

let S be non empty non void n PC-correct PCLangSignature ; :: thesis: for L being language MSAlgebra over S

for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds

(A \or C) \imp (B \or D) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds

(A \or C) \imp (B \or D) in F

let F be PC-theory of L; :: thesis: for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds

(A \or C) \imp (B \or D) in F

let A, B, C, D be Formula of L; :: thesis: ( A \imp B in F & C \imp D in F implies (A \or C) \imp (B \or D) in F )

assume A1: A \imp B in F ; :: thesis: ( not C \imp D in F or (A \or C) \imp (B \or D) in F )

assume A2: C \imp D in F ; :: thesis: (A \or C) \imp (B \or D) in F

A3: (A \imp (B \or D)) \imp ((C \imp (B \or D)) \imp ((A \or C) \imp (B \or D))) in F by Def38;

( B \imp (B \or D) in F & D \imp (B \or D) in F ) by Def38;

then A4: ( A \imp (B \or D) in F & C \imp (B \or D) in F ) by A1, A2, Th45;

then (C \imp (B \or D)) \imp ((A \or C) \imp (B \or D)) in F by A3, Def38;

hence (A \or C) \imp (B \or D) in F by A4, Def38; :: thesis: verum

for L being language MSAlgebra over S

for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds

(A \or C) \imp (B \or D) in F

let S be non empty non void n PC-correct PCLangSignature ; :: thesis: for L being language MSAlgebra over S

for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds

(A \or C) \imp (B \or D) in F

let L be language MSAlgebra over S; :: thesis: for F being PC-theory of L

for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds

(A \or C) \imp (B \or D) in F

let F be PC-theory of L; :: thesis: for A, B, C, D being Formula of L st A \imp B in F & C \imp D in F holds

(A \or C) \imp (B \or D) in F

let A, B, C, D be Formula of L; :: thesis: ( A \imp B in F & C \imp D in F implies (A \or C) \imp (B \or D) in F )

assume A1: A \imp B in F ; :: thesis: ( not C \imp D in F or (A \or C) \imp (B \or D) in F )

assume A2: C \imp D in F ; :: thesis: (A \or C) \imp (B \or D) in F

A3: (A \imp (B \or D)) \imp ((C \imp (B \or D)) \imp ((A \or C) \imp (B \or D))) in F by Def38;

( B \imp (B \or D) in F & D \imp (B \or D) in F ) by Def38;

then A4: ( A \imp (B \or D) in F & C \imp (B \or D) in F ) by A1, A2, Th45;

then (C \imp (B \or D)) \imp ((A \or C) \imp (B \or D)) in F by A3, Def38;

hence (A \or C) \imp (B \or D) in F by A4, Def38; :: thesis: verum